HW10sol_FA08 - UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN...

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Unformatted text preview: UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN Department of Electrical and Computer Engineering ECE 410 Digital Signal Processing Homework 10 Solution Prof. Bresler, Prof. Jones Friday, November 7, 2008 Problem 1 (15 points) A speech signal x a ( t ) is assumed to be a bandlimited to 12kHz. It is desired to filter this signal with a bandpass filter that will pass the frequencies between 300 Hz to 6kHz by using a digital filter H d ( ω ) sandwiched between an A/D and an ideal D/A. (a) Determine the Nyquist sampling rate for the input signal. T Nyquist = 1 (12)(10 3 )(2) = 0 . 000042sec = 0 . 042 ms ⇒ f Nyquist = 1 /T Nyquist = 24 kHz (b) Sketch the frequency response H d, 1 ( ω ) for the necessary discrete-time filter, when sampling at the Nyquist rate. ω max = 1 (12)(10 3 )(2) (2 π )(6000) = π/ 2 ω min = 1 (12)(10 3 )(2) (2 π )(300) = π/ 40 Figure 1: Problem 1(b) solution: Frequency response of the filter, when sampling at the Nyquist rate (c) Find the largest sampling period T for which the A/D, digital filter response ( H d, 2 ( ω )), and D/A can perform the desired filtering function. (Hint: Some amount of aliasing may be permissible during A/D conversion for this part.) Some aliasing of the input signal is allowed with the condition that the minimum alaising frequency is greater then the cutoff frequency of the filter. 1 2 π- 2 π · 12000 T max 1 ≥ 2 π · 6000 T max 1 1 ≥ 18000 T max 1 T max 1 = 1 18000 sec = 5 . 556 × 10- 5 sec Also, the maximum frequency of H d, 1 ( ω ) should be less than or equal to π . ω max = T max 2 (2 π )(6000) = π T max 2 = 1 12000 sec. Therefore, we take T = min( T max 1 ,T max 2 ) = T max 1 = 5 . 556 × 10- 5 sec (d) For the system using T from part (c), sketch the necessary H d ( ω ). ω max = (1 / 18000 s )(6000)(2 π ) = 3 π/ 2 ω min = ω max = (1 / 18000 s )(300)(2 π ) = π/ 30 Figure 2: Problem 1(d) solution: Frequency response of the filter, when sampling at the maximum rate Problem 2 (10 points) The transfer functions of three LSI systems are given below. For each system, determine if it is a FIR or an IIR filter. (a) H ( z ) = z 2 +8 z +12 z +6 H ( z ) = z +6 ( z +6)( z +2) = z + 2 Since H ( z ) can be written as a polynomial it is a FIR filter. (b) H ( z ) = z +6 z 2 +8 z +12 H ( z ) = z +6 ( z +6)( z +2) = 1 z +2 Since H ( z ) cannot be written as a polynomial it is a IIR filter. (c) H ( z ) = z 2 + 3 z Since H ( z ) can be written as a polynomial it is a FIR filter. Problem 3 (15 points) 2 The frequency response of a GLP filter can be expressed as H d ( ω ) = R ( ω ) e j ( α- Mω ) where R ( ω ) is a real function. For each of the following filters, determine whether it is a GLP filter. If it is, find R ( ω ), M , and α , and indicate whether it is also a linear phase filter....
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HW10sol_FA08 - UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN...

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